Sufficient condition for universal quantum computation using bosonic circuits

TL;DR

This paper establishes a sufficient condition for universal quantum computation in continuous-variable bosonic circuits by mapping states to qubits and analyzing resourcefulness, expanding understanding of what enables quantum computational universality.

Contribution

The paper introduces a general framework for mapping continuous-variable states to qubits and formulates a sufficient condition for universality in bosonic circuits, advancing resource theory in quantum computation.

Findings

Formulated a sufficient condition for universal quantum computation.

Evaluated resourcefulness of Gaussian, GKP, and cat states.

Demonstrated that certain decompositions can be constructed from simulatable operations.

Abstract

Continuous-variable bosonic systems stand as prominent candidates for implementing quantum computational tasks. While various necessary criteria have been established to assess their resourcefulness, sufficient conditions have remained elusive. We address this gap by focusing on promoting circuits that are otherwise simulatable to computational universality. The class of simulatable, albeit non-Gaussian, circuits that we consider is composed of Gottesman-Kitaev-Preskill (GKP) states, Gaussian operations, and homodyne measurements. Based on these circuits, we first introduce a general framework for mapping a continuous-variable state into a qubit state. Subsequently, we cast existing maps into this framework, including the modular and stabilizer subsystem decompositions. By combining these findings with established results for discrete-variable systems, we formulate a sufficient condition for achieving universal quantum computation. Leveraging this, we evaluate the computational resourcefulness of a variety of states, including Gaussian states, finite-squeezing GKP states, and cat states. Furthermore, our framework reveals that both the stabilizer subsystem decomposition and the modular subsystem decomposition (of position-symmetric states) can be constructed in terms of simulatable operations. This establishes a robust resource-theoretical foundation for employing these techniques to evaluate the logical content of a generic continuous-variable state, which can be of independent interest.